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A heterogeneous mean-field SIRS infectious disease stochastic dynamic model with population size is established, the main period of the model and the average extinction time of the disease are obtained, and the critical community size threshold of the model is obtained by using the relationship between the main period and average extinction time. The results show that: when average degrees of networks are equal but degree distributions differ, or vice versa, the thresholds for critical community size vary significantly. Population size is an important parameter affecting the extinction of stochastic infectious diseases, and this result can provide a theoretical basis for the control and eradication of infectious diseases. [ABSTRACT FROM AUTHOR]

The paper discusses the Galton–Watson stochastic branching system. We deal only with the noncritical case. Our task in the work is to improve our recent result which explicitly calculated the famous constant in the theory of subcritical Galton–Watson branching systems, announced by Kolmogorov in 1938. We demonstrate the rate of convergence to the Kolmogorov constant. This improvement contributes to determining the speed of approximation rate in a number of classical limit theorems of the theory of Galton–Watson branching systems. [ABSTRACT FROM AUTHOR]

Extensive studies have been conducted on effect of sealing ratio on combustion process and temperature field of pool fire, while still lacking emergency rescue research after the failure of the original fire extinguishing system. Therefore, a series of tests were carried out in a 1/3 reduced-scale tunnel to explore the nitrogen injection into the closed tunnel on the effectiveness of fire-fighting. The results show that, the presence of nitrogen not only significantly lowers the oxygen concentration and mass loss rate of the air drawn into the combustion area, but the flame continues to entrap the polluted low-oxygen air, resulting in the continuous inadequate combustion of the fire and a large amount of asphyxiating smoke, thus shortening the self-extinguishing time. The concentration range of oxygen concentration of flame extinguishing during nitrogen injection is 11.13 %–14.98 %. Simultaneously, an analytically derived model to estimate the extinction time of nitrogen injection into the closed utility tunnel was developed, which is found that the deviation between the improved classical model and the experimental data within 10 %. The present work positively provides a tool for the assessment of thermal environment and estimation of fire detection time in engineering practice.

A massive vaccination programme against COVID-19 infection started at the beginning of 2021. Studies show that vaccinated people are subject to reinfection, and there is uncertainty in the rate of immunity loss, the force of infection, recovery rate and vaccine efficacy. Here we study a six-dimensional stochastic epidemic model with vaccine-induced immunity loss to demonstrate the effect of vaccination in controlling the COVID-19 epidemic. It is shown that the disease persists for a long time if the stochastic basic reproduction number R 0 V S > 1 holds. We have also proved a sufficient condition for disease eradication. Our analysis shows that the disease cannot persist if R 0 V ext < 1 . However, this latter condition may not hold if the infectivity increases and/or the vaccine-induced immunity loss increases. Indian and Italian COVID-19 data are used to demonstrate various dynamical behaviours of the system and disease persistence. A non-trivial observation is that mass vaccination cannot eradicate the disease if the vaccine-induced immunity loss is high. Disease eradication is also challenging with the ongoing immunization process if the infectivity of the virus is also high. These results decipher that the infection will last long unless a long-lasting vaccine candidate appears or a low infectious variant replaces the highly contagious COVID-19 variant. [ABSTRACT FROM AUTHOR]

The article compares different versions of individual-based models of single population dynamics with overlapping generations with results of the model of population with non-overlapping generations. In all models, various versions of global competition for common resources and their unequal partitioning between competing individuals are analysed, i.e., when juveniles and adult individuals compete for the same resources, when juveniles and adult individuals use different resources, and a case called mother’s care. The article analyses the relationship between individual variability and population persistence measured by the time of population extinction. Persistence of a population is increased by all forms of competition between individuals not weakened by any environmental factors (e.g. diversification of resources of competing individuals) or factors arising from evolutionary history of the species (e.g. mother’s care) and related in an appropriate way to the mechanism shaping individual variability in the population. Any form of weakening competition leading to decreasing individual variability will adversely affect the persistence of the population. However, differences between versions of the model are not very big.

The Scottish Small Isles – comprising Muck, Rùm, Eigg, Canna, and by extension, Coll – are geologically complex, with intersecting rock samples from the Archean (Lewisian Gneiss basements formed approximately 3 billion years ago), Proterozoic (Torridonian sandstone formed approximately 1 billion years ago), Mesozoic (sedimentation deposited approximately 200 million years ago) and Palaeocene (basalt formed approximately 55.8 million years ago as part of the Palaeocene–Eocene Thermal Maximum event). This practice research article – drawing on palaeontology, kinaesthetic learning and creative writing – takes the Small Isles as a case study for what geologist Marcia Bjornerud defines as a discernible “timefulness” that humans should seek to attain: “an acute consciousness of how the world is made by–indeed, made of–time” (2020, Timefulness: How Thinking Like a Geologist Can Help Save the World, 5). Through their lithic intrusions, and interruptive strata, the Small Isles offer an alternative form of pedagogy: where multiple epochs, tenses and tempos visibly converse with one another; where “polytemporality” can be witnessed and physically experienced; where the notion of linear time is destabilised.

The first goal of this paper is to establish a result on the existence and uniqueness of solution to an initial-boundary value problem for parabolic equations of Kirchhoff type involving the 1-Laplace operator. The second goal is to discuss some qualitative properties, such as asymptotic behaviour and the extinction of solutions for the considered problem. [ABSTRACT FROM AUTHOR]

We study the Cauchy-Dirichlet Problem (CDP) for a nonlinear and nonlocal diffusion equation of singular type of the form $ \partial_t u = - \mathcal{L} u^m $ posed on a bounded Euclidean domain $ \Omega\subset \mathbb{R}^N $ with smooth boundary and $ N\ge 1 $. The linear diffusion operator $ \mathcal{L} $ is a sub-Markovian operator, allowed to be of nonlocal type, while the nonlinearity is of singular type, namely $ u^m = |u|^{m-1}u $ with $ 0

Neutral models in ecology assume that all species are demographically equivalent, such that their abundances differ ultimately because of demographic stochasticity rather than selection. In spite of their simplicity, neutral models have been found to accurately reproduce static patterns of biodiversity for diverse communities. However, the same neutral models have been found to exhibit species abundance dynamics that are far too slow compared to reality, resulting in poor fits to temporally dynamic patterns of biodiversity. Here, we show that one of the root causes of these slow dynamics is the additional assumption that a community has reached an equilibrium with a fixed community size, with species that have a net growth rate close to zero. We removed this additional assumption by constructing and analyzing a neutral model with an expected community size that can change over time and is not necessarily at equilibrium, which thus allows the historical formation of a community to be represented explicitly. Our analysis demonstrated that for the general scenario where a small community rapidly grows in size to a carrying capacity, representing recovery from ecological disturbance or assembly of a new community, the model produced much larger changes in species abundances and much shorter species ages than a neutral model at an equilibrium with fixed community size. In addition, the species abundance distribution was biphasic with a subset of abundant species arising from a founder effect. We confirmed these new results in applications of the new model to the specific scenario of recovery of the Amazon tree community after the end-Cretaceous bolide impact, which involved periods of increasing and decreasing community size. We conclude that incorporating transient dynamics in neutral models improves realism by allowing explicit consideration of how a community is formed over realistic time-scales. • Transient dynamics of a neutral model with varying community size are analyzed. • Formulae are derived specifying expected species richness and species age. • Formulae are derived specifying the species-abundance distribution. • Model gives a biphasic SAD with common species arising from a founder effect. • Model also gives realistic species ages for a tree community at an Amazon scale. [ABSTRACT FROM AUTHOR]

We introduce a class of branching processes in which the reproduction or lifetime distribution at a given time depends on the total cumulative number of individuals who have been born in the population until that time. We focus on a continuous-time version of these processes, called total-progeny-dependent birth-and-death processes, and study some of their properties through the analysis of their deterministic (fluid) approximation. These properties include the maximum population size, the total progeny size at extinction, the time to reach the maximum population size, and the time until extinction. As the fluid approximation does not allow us to determine the time until extinction directly, we propose several methods to complement this approach. We also use the deterministic approach to study the behavior of the processes as we increase the magnitude of the individual's birth rate. [ABSTRACT FROM AUTHOR]

In this study, droplet tests were carried out for pure and blended forms of diesel fuel, which is the traditional fuel, and trimethyl borate fuel, which is new generation fuels. In this context, the evolution of fuel droplet diameter, flame structure, and flame temperature over time was observed using a high-speed camera and a thermal camera simultaneously. Measurements were made for the addition of 20%, 40%, 60%, and 80% trimethyl borate fuel to diesel fuel in fuel blends. The curves of the time-dependent change of the dimensionless square of the droplet diameter (D/D 0)2 of the fuel droplets considered in the study, in general, have been determined to comply with the D2-law. Also, as the amount of diesel rose, there were high changes in droplet distortion and high deviations from the hemispherical geometry, and also elongation in the dominant shape droplet shape changed times. While the highest flame temperature was monitored by the thermal camera in the flame formed by the trimethyl borate droplet, as the amount of diesel fuel in the mixture increased, the maximum flame temperature decreased, but the burning time of the droplet was prolonged. On the other hand, the shortest ignition delay was measured for trimethyl borate, while the longest ignition delay time was detected for diesel fuel droplet. In general, it has been observed that the addition of trimethyl borate reduces the ignition delay, shortens the extinction time, and increases the temperature during combustion compared to diesel. • The burning time was prolonged as the amount of diesel in the blend increased. • The highest flame temperature was observed for the trimethyl borate droplet. • As the amount of diesel in the blend raised, the flame temperature decreased. • The shortest ignition delay was measured for trimethyl borate droplet. • As the amount of trimethyl borate raised, droplet deformation was upsurged. [ABSTRACT FROM AUTHOR]

In this paper we introduce new birth-and-death processes with partial catastrophe and study some of their properties. In particular, we obtain some estimates for the mean catastrophe time, and the first and second moments of the distribution of the process at a fixed time t. This is completed by some asymptotic results. [ABSTRACT FROM AUTHOR]

Abstract Continuum limits in the form of stochastic differential equations are typically used in theoretical population genetics to account for genetic drift or more generally, inherent randomness of the model. In evolutionary game theory and theoretical ecology, however, this method is used less frequently to study demographic stochasticity. Here, we review the use of continuum limits in ecology and evolution. Starting with an individual‐based model, we derive a large population size limit, a (stochastic) differential equation which is called continuum limit. By example of the Wright–Fisher diffusion, we outline how to compute the stationary distribution, the fixation probability of a certain type, and the mean extinction time using the continuum limit. In the context of the logistic growth equation, we approximate the quasi‐stationary distribution in a finite population.

Consider two-type linear-fractional branching processes in varying environments with asymptotically constant mean matrices. Let $\nu$ be the extinction time. Under certain conditions, we show that both $\mathbb{P}(\nu=n)$ and $\mathbb{P}(\nu>n)$ are asymptotically the same as some functions of the products of spectral radii of the mean matrices. We also give an example for which $\mathbb{P}(\nu=n)$ decays with various speeds such as ${c}/({n^{1/2}\log n)^2}$ , ${c}/{n^\beta}$ , $\beta >1$ , which are very different from those of homogeneous multitype Galton–Watson processes. [ABSTRACT FROM AUTHOR]

We study the property of extinction in a finite time for nonnegative solutions of 1 q ∂ ∂ t (u q) − ∇ (| ∇ u | p − 2 ∇ u) + a (x) u λ = 0 for the Dirichlet Boundary Conditions when q > λ > 0 , p ⩾ 1 + q , p ⩾ 2 , a (x) ⩾ 0 and Ω a bounded domain of R N (N ⩾ 1). We prove some necessary and sufficient conditions. The threshold is for power functions when p > 1 + q while finite time extinction occurs for very flat potentials a (x) when p = 1 + q. [ABSTRACT FROM AUTHOR]

Attalea nucifera is an acaulescent palm native of Colombia that is in risk of extinction. Between 2016-2017 we evaluated the structure and density of populations in six localities of the Magdalena river val-ley, and studied the population dynamic in the locality of Guaduas, Cundinamarca (VC). Although the population structure differs among locations (X2 = 1819; gl = 25; P < 0.05), populations tend to group according to the degree of habitat perturbation. In four localities an inverted J population structure was observed. A matrix population model showed a finite growth rate (λ) of 0.979 (CI95 % = 0.962–0.997). The demographic processes of the permanence of seedlings, sub-adults and young adults show more elasticity. A transient dynamic simulation projected to 30 years shows that under the scenarios of paddock and cattle lopping the population size decrease drastically. The extinction threshold calcu-lated for the population in the locality VC is of 145 years, but paddock and cattle lopping activities can reduce it to less than 40 years. Although Attalea nucifera persists in very disturbing locations in the Middle Magdalena Basin, the results of population dynamics in the locality VC suggest that it could be less tolerant of environmental disturbances. Thus, it is necessary to increase our knowledge of its population dynamics, as well as seed germination and seedling establishment in different disturbance conditions.

Abstract Background Red Queen dynamics are defined as long term co-evolutionary dynamics, often with oscillations of genotype abundances driven by fluctuating selection in host-parasite systems. Much of our current understanding of these dynamics is based on theoretical concepts explored in mathematical models that are mostly (i) deterministic, inferring an infinite population size and (ii) evolutionary, thus ecological interactions that change population sizes are excluded. Here, we recall the different mathematical approaches used in the current literature on Red Queen dynamics. We then compare models from game theory (evo) and classical theoretical ecology models (eco-evo), that are all derived from individual interactions and are thus intrinsically stochastic. We assess the influence of this stochasticity through the time to the first loss of a genotype within a host or parasite population. Results The time until the first genotype is lost (“extinction time”), is shorter when ecological dynamics, in the form of a changing population size, is considered. Furthermore, when individuals compete only locally with other individuals extinction is even faster. On the other hand, evolutionary models with a fixed population size and competition on the scale of the whole population prolong extinction and therefore stabilise the oscillations. The stabilising properties of intra-specific competitions become stronger when population size is increased and the deterministic part of the dynamics gain influence. In general, the loss of genotype diversity can be counteracted with mutations (or recombination), which then allow the populations to recurrently undergo negative frequency-dependent selection dynamics and selective sweeps. Conclusion Although the models we investigated are equal in their biological motivation and interpretation, they have diverging mathematical properties both in the derived deterministic dynamics and the derived stochastic dynamics. We find that models that do not consider intraspecific competition and that include ecological dynamics by letting the population size vary, lose genotypes – and thus Red Queen oscillations – faster than models with competition and a fixed population size.

Given a phylogenetic tree that includes only extinct, or a mix of extinct and extant taxa, where at least some fossil data are available, we present a method to compute the distribution of the extinction time of a given set of taxa under the Fossilized-Birth-Death model. Our approach differs from the previous ones in that it takes into account (i) the possibility that the taxa or the clade considered may diversify before going extinct and (ii) the whole phylogenetic tree to estimate extinction times, whilst previous methods do not consider the diversification process and deal with each branch independently. Because of this, our method can estimate extinction times of lineages represented by a single fossil, provided that they belong to a clade that includes other fossil occurrences. We assess and compare our new approach with a standard previous one using simulated data. Results show that our method provides more accurate confidence intervals. This new approach is applied to the study of the extinction time of three Permo-Carboniferous synapsid taxa (Ophiacodontidae, Edaphosauridae, and Sphenacodontidae) that are thought to have disappeared toward the end of the Cisuralian (early Permian), or possibly shortly thereafter. The timing of extinctions of these three taxa and of their component lineages supports the idea that the biological crisis in the late Kungurian/early Roadian consisted of a progressive decline in biodiversity throughout the Kungurian. [ABSTRACT FROM AUTHOR]

An individual-based approach is used to describe population dynamics. Two kinds of models have been constructed with different distributions illustrating individual variability. In both models, the growth rate of an individual and its final body weight at the end of the growth period, which determines the number of offspring, are functions of the amount of resources assimilated by an individual. In the model with a symmetric distribution, the half saturation constant in the Michaelis–Menten function describing the relationship between the growth of individuals and the amount of resources has a normal distribution. In the model with an asymmetric distribution, resources are not equally partitioned among individuals. The individual who acquired more resources in the past, will acquire more resources in the future. A single population comprising identical individuals has a very short extinction time. If individuals differ in the amount of food assimilated, this time significantly increases irrespectively of the type of model describing population dynamics. Individuals of two populations of competing species use common resources. For larger differences in individual variability, the more variable species will have a longer extinction time and will exclude less variable species. Both populations can also coexist when their variabilities are equal or even when they are slightly different, in the latter case under the condition of high variability of both species. These conclusions have a deterministic nature in the case of the model with the asymmetric distribution—repeated simulations give the same results. In the case of the model with the symmetric distribution, these conclusions are of a statistical nature—if we repeat the simulation many times, then the more variable species will have a longer extinction time more frequently, but some results will happen (although less often) when the less variable species has a longer extinction time. Additionally, in the model with the asymmetric distribution, the result of competition will depend on the way of the introduction of variability into the model. If the higher variability is due to an increase in the proportion of individuals with a low assimilation of resources, it can produce a longer extinction time of the less variable species. [ABSTRACT FROM AUTHOR]

Given a phylogenetic tree that includes only extinct, or a mix of extinct and extant taxa, where at least some fossil data are available, we present a method to compute the distribution of the extinction time of a given set of taxa under the Fossilized-Birth-Death model. Our approach differs from the previous ones in that it takes into account (i) the possibility that the taxa or the clade considered may diversify before going extinct and (ii) the whole phylogenetic tree to estimate extinction times, whilst previous methods do not consider the diversification process and deal with each branch independently. Because of this, our method can estimate extinction times of lineages represented by a single fossil, provided that they belong to a clade that includes other fossil occurrences. We assess and compare our new approach with a standard previous one using simulated data. Results show that our method provides more accurate confidence intervals. This new approach is applied to the study of the extinction time of three Permo-Carboniferous synapsid taxa (Ophiacodontidae, Edaphosauridae, and Sphenacodontidae) that are thought to have disappeared toward the end of the Cisuralian (early Permian), or possibly shortly thereafter. The timing of extinctions of these three taxa and of their component lineages supports the idea that the biological crisis in the late Kungurian/early Roadian consisted of a progressive decline in biodiversity throughout the Kungurian.

Water mist based systems have emerged as potentially lucrative alternatives to Halon-based fire suppression systems and have been extensively studied in the literature. In the present work, experimental investigations are carried out on the performance of the water mist nozzles in extinguishing pool fires of diameter 0.1 m, 0.2 m, and 0.3 m with diesel and heptane as fuels. The growth of the fire is studied for these fires by measuring the transient mass burning rates. Water mist is generated by passing plain water at room temperature through single-head solid-cone pressure-swirl nozzles with X-type swirl-inserts. The nozzle is directed vertically downwards and operated at various heights (0.75 m to 2 m) from the fuel surface with the injection pressure ranging from 6 bar to 10 bar. The extinction performances of four nozzles with orifice diameters of 1.2 mm, 1.65 mm, 1.9 mm, and 2.45 mm are investigated through videography. A comparison of the extinction performances by all the nozzles is presented and the optimal nozzle based on the consumption of water is reported for a given fire. An empirical correlation for the extinction time is reported in both the dimensional and non-dimensional form. Observed interactions of the spray with fuel surface are reported depicting probable undesired phenomena in the application of the water-mist system with top injection. [ABSTRACT FROM AUTHOR]

We consider branching Brownian motion on the real line with absorption at zero, in which particles move according to independent Brownian motions with the critical drift of -√2. Kesten (Stoch Process 7:9–47, 1978) showed that almost surely this process eventually dies out. Here we obtain upper and lower bounds on the probability that the process survives until some large time t. These bounds improve upon results of Kesten (Stoch Process 7:9–47, 1978), and partially confirm nonrigorous predictions of Derrida and Simon (EPL 78:60006, 2007).

Continuum limits in the form of stochastic differential equations are typically used in theoretical population genetics to account for genetic drift or more generally, inherent randomness of the model. In evolutionary game theory and theoretical ecology, however, this method is used less frequently to study demographic stochasticity. Here, we review the use of continuum limits in ecology and evolution. Starting with an individual‐based model, we derive a large population size limit, a (stochastic) differential equation which is called continuum limit. By example of the Wright–Fisher diffusion, we outline how to compute the stationary distribution, the fixation probability of a certain type, and the mean extinction time using the continuum limit. In the context of the logistic growth equation, we approximate the quasi‐stationary distribution in a finite population. [ABSTRACT FROM AUTHOR]

The spatial arrangement of habitat patches in a metapopulation and the dispersal connections among them influence metapopulation persistence. Metapopulation persistence emerges from a dynamic process, namely the serial extinctions and recolonizations of local habitat patches, while measures of persistence are typically based solely on structural properties of the spatial network (e.g., spatial distance between sites). Persistence estimators based on static properties may be unable to capture the dynamic nature of persistence. Understanding the shape of the distribution of extinction times is a central goal in population ecology. Here, we examine the goodness of fit of the power law to patch persistence time distributions using data on a foundational metapopulation system—the Glanville fritillary butterfly in the Åland islands. Further, we address the relationship between structural measures of metapopulation persistence (i.e., metapopulation capacity) and our temporal distributional fits to patch persistence times based on a power law. Patch persistence time distributions were well fit by a power law for the majority of semi-independent networks. Power law fits to persistence time distributions were related to metapopulation capacity, linking structural and temporal measures of metapopulation persistence. Several environmental variables and measures of network topology were correlated with both measures of metapopulation persistence, though correlations tended to be stronger for the structural measure of metapopulation persistence (i.e., metapopulation capacity). Together, our findings suggest that persistence time distributions are useful dynamic properties of metapopulations, and provide evidence of a relationship between metapopulation structure and metapopulation dynamics. [ABSTRACT FROM AUTHOR]

How long a newly emerging trait will stay in a population is a fundamental but rarely asked question in cultural evolution. To tackle this question, the distribution and mean of the time to extinction of a discrete cultural trait are derived for models with overlapping generations, in which trait transmission occurs from multiple role models to a single newborn and may fail with a certain probability. We explore two models. The first is a Moran-type model, which allows us to derive the exact analytical formula for the mean time to extinction of a trait in a finite population. The second is a branching process, which assumes an infinitely large population and allows us to derive approximate analytical formulae for the distribution and mean of the time to extinction in the first model under a large population size. We show that in the first model, the mean time to extinction apparently diverges (becomes so large that even numerical computation is impractical) under a certain parameter condition as the population size tends to infinity. Using the second model, we explain the underlying mechanism of the apparent divergence found in the first model and derive the mathematical condition for this divergence in terms of transmission efficiency and the number of role models per newborn. When this mathematical condition is satisfied in the second model, the probability of extinction is less than 1, and the mean extinction time does not exist. In addition, we find that in both models, the time to extinction of the trait becomes longer as the number of role models per individual increases and as cultural transmission becomes more efficient. • Extinction time of a cultural trait is computed for a finite-population model. • Extinction time apparently diverges for certain parameter values. • Approximations are derived for extinction time using a branching process. • Divergence of extinction time is explained in terms of phase transition theory. [ABSTRACT FROM AUTHOR]

We consider a continuous-time stochastic model of spiking neurons originally introduced by Ferrari et al. in Ferrari et al. (2018). In this model, we have a finite or countable number of neurons which are vertices in some graph G where the edges indicate the synaptic connection between them. We focus on metastability, understood as the property for the time of extinction of the network to be asymptotically memory-less, and we prove that this model exhibits two different behaviors depending on the nature of the specific underlying graph of interaction G that is chosen. In this model the spiking activity of any given neuron is represented by a point process, whose rate fluctuates between 1 and 0 over time depending on whether the membrane potential is positive or null. The membrane potential of each neuron evolves in time by integrating all the spikes of its adjacent neurons up to the last spike of the said neuron, so that when a neuron spikes, its membrane potential is reset to 0 while the membrane potential of each of its adjacent neurons is increased by one unit. Moreover, each neuron is exposed to a leakage effect, modeled as an abrupt loss of membrane potential which occurs at random times driven by a Poisson process of some fixed rate γ. It was previously proven that when the graph G is the infinite one-dimensional lattice, this model presents a phase transition with respect to the parameter γ. It was also proven that, when γ is small enough, the renormalized time of extinction (the first time at which all neurons have a null membrane potential) of a finite version of the system converges in law toward an exponential random variable when the number of neurons goes to infinity. The present article is divided into two parts. First we prove that, in the finite one-dimensional lattice, this last result does not hold anymore if γ is large enough, and in fact we prove that for γ > 1 the renormalized time of extinction is asymptotically deterministic. Then we prove that conversely, if G is the complete graph, the result of metastability holds for any positive γ. [ABSTRACT FROM AUTHOR]

Consider a general continuous-state branching process with additional interaction, which destroys the branching property. We give precise conditions on the interaction term, in order to decide whether the extinction time of the process remains or not bounded as the initial value tends to infinity, and similarly for the total mass of the process. [ABSTRACT FROM AUTHOR]

This work is concerned with the gradient flow of absolutely p-homogeneous convex functionals on a Hilbert space, which we show to exhibit finite ( p < 2 ) or infinite extinction time ( p ≥ 2 ). We give upper bounds for the finite extinction time and establish sharp convergence rates of the flow. Moreover, we study next order asymptotics and prove that asymptotic profiles of the solution are eigenfunctions of the subdifferential operator of the functional. To this end, we compare with solutions of an ordinary differential equation which describes the evolution of eigenfunction under the flow. Our work applies, for instance, to local and non-local versions of PDEs like p-Laplacian evolution equations, the porous medium equation, and fast diffusion equations, herewith generalizing many results from the literature to an abstract setting. We also demonstrate how our theory extends to general homogeneous evolution equations which are not necessarily a gradient flow. Here, we discover an interesting integrability condition which characterizes whether or not asymptotic profiles are eigenfunctions. [ABSTRACT FROM AUTHOR]

Summary: The extinguishing performance of three dry chemical powders (DCPs) was investigated through a lab‐scale suppression system for liquefied petroleum gas (LPG) fire. The magnesium hydroxide (Mg(OH)2), sodium bicarbonate (NaHCO3), and commercial ABC − MAP powders are used to prepare two groups of samples: raw samples and milled samples. The effect of milling action on the properties of DCPs, such as the bulk density, microstructure, particle size, thermal decomposition, and the extinguishing performance, is analyzed. The density test revealed that the bulk density increased after milling, and the Mg(OH)2 had a lower density than the other powders. The microstructure analysis showed that the milled powders had a smaller particle size and more regular shapes than raw powders. The thermal analysis demonstrated that the powder decomposition process was somewhat similar with a slight difference in initial decomposition temperature and degradation rate. The fire tests proved that the extinguishing efficacy greatly improved after the samples were milled. Moreover, the experiments indicated that the milled Mg(OH)2 was superior in fire extinguishing to the other samples with shorter extinction time and less agent quantity consumed. Based on the results, it can be inferred that the milling action has a significant influence in enhancing the extinguishing mechanisms' efficiency of DCPs. [ABSTRACT FROM AUTHOR]

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Frequency-dependent selection reflects the interaction between different species as they battle for limited resources in their environment. In a stochastic evolutionary game the species relative fitnesses guides the evolutionary dynamics with fluctuations due to random drift. A selection advantage which depends on a changing environment will introduce additional possibilities for the dynamics. We analyse a simple model in which a random environment allows competing species to coexist for a long time before a fixation of a single species happens. In our analysis we use stability in a linear combination of competing species to approximate the stochastic dynamics of the system by a diffusion on a one dimensional co-existence region. Our method significantly simplifies approximating the probability of first extinction and its expected time, and demonstrates a rigorous model reduction technique for evaluating quasistationary properties of stochastic evolutionary dynamics. [ABSTRACT FROM AUTHOR]

Background: Red Queen dynamics are defined as long term co-evolutionary dynamics, often with oscillations of genotype abundances driven by fluctuating selection in host-parasite systems. Much of our current understanding of these dynamics is based on theoretical concepts explored in mathematical models that are mostly (i) deterministic, inferring an infinite population size and (ii) evolutionary, thus ecological interactions that change population sizes are excluded. Here, we recall the different mathematical approaches used in the current literature on Red Queen dynamics. We then compare models from game theory (evo) and classical theoretical ecology models (eco-evo), that are all derived from individual interactions and are thus intrinsically stochastic. We assess the influence of this stochasticity through the time to the first loss of a genotype within a host or parasite population. Results: The time until the first genotype is lost ("extinction time"), is shorter when ecological dynamics, in the form of a changing population size, is considered. Furthermore, when individuals compete only locally with other individuals extinction is even faster. On the other hand, evolutionary models with a fixed population size and competition on the scale of the whole population prolong extinction and therefore stabilise the oscillations. The stabilising properties of intra-specific competitions become stronger when population size is increased and the deterministic part of the dynamics gain influence. In general, the loss of genotype diversity can be counteracted with mutations (or recombination), which then allow the populations to recurrently undergo negative frequency-dependent selection dynamics and selective sweeps. Conclusion: Although the models we investigated are equal in their biological motivation and interpretation, they have diverging mathematical properties both in the derived deterministic dynamics and the derived stochastic dynamics. We find that models that do not consider intraspecific competition and that include ecological dynamics by letting the population size vary, lose genotypes – and thus Red Queen oscillations – faster than models with competition and a fixed population size. [ABSTRACT FROM AUTHOR]